If the outer boundary of the
compact plane set E is the union of finitely many disjoint analytic Jordan curves, the
Garabedian function of E is a familiar object. J. Garnett and S. Y. Havinson have
each asked whether the Garabedian functions of a decreasing sequence of such sets
must converge. The present paper shows that they do converge. This fact leads to a
natural definition of the Garabedian function of an arbitrary compact plane set. As
an intermediate step, an approximate formula is obtained for the analytic
capacity of the union of a compact set E and a small disc not intersecting
E.
